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混合線形モデルにおける分散成分の平均情報アルゴリズムによる制限最尤推定に関する研究
混合線形モデルにおける分散成分の平均情報アルゴリズムによる制限最尤推定に関する研究
蘆田, 一郎
The method for estimating variance components in linear models is one of the most important statistical methods and is used in various fields of researches and applications. Especially in the field of animal breeding and genetics where economically important quantitative traits are treated, accurate estimates of the variance components are required in estimating genetic parameters, making inferences about the genetic merit of an animal or population of animals, and constructing a breeding plan. Of various methods for variance component estimation developed until now, the representative of the certain best methods is the restricted maximum likelihood (REML) procedure. The REML procedure has been used as the official method in genetic evaluation systems in the world. On REML, however, further investigation is required to reveal the detailed properties of the sampling (co) variances of the estimates, since REML is an iterative method. Moreover, it is necessary to develop a more efficient computing algorithm for REML estimation of variance components in the general mixed linear model from unbalanced and large data sets. The present study was undertaken to find an effective expression for the sampling covariance matrix of the REML estimators, noticing the minimum variance quadratic unbiased estimation (MIVQUE) of variance components, and to develop a new efficient numerical technique that can handle the whole process of the point estimation and the sampling (co) variance estimation. I. The MIVQUE based-sampling covariance matrix of the REML estimators Previously, the sampling (co) variances of the REML estimators defined as the elements of the inverse matrix of the Fisher information matrix, which is represent the large sample asymptotic sampling (co) variances. Herein, noticing the structure of the sampling (co) variances of the MIVQUE estimators whose prior information is the REML estimators and assuming the basic mixed linear model, an expression for the sampling covariance matrix of the REML estimators was derived. The resulting elements of the derived covariance matrix (denoted as the V_1 matrix) included the traces of huge products of the inverse of the variance matrix of the random vector and a partitioned matrix of a g-inverse of the mixed model coefficient matrix, the true components of variances and their REML estimators. A Monte Carlo simulation conducted indicated that for a limited size of the sample, use of the derived V_1 matrix is more valid than that of the inverse of the Fisher information matrix, when true variances are known. However, the large computational requirement with the V_1 matrix suggested the necessity of finding its more simplified form.
II. A sampling covariance matrix of the REML estimators derived by the newly proposed pseudo-variance approach To reduce the computational load in the calculation of the V_1 matrix, an approach which will be called the pseudo-variance approach (PVA) was proposed, introducing the large sample theory. The term ‘pseudo-variance’ means deriving variances of quadratic forms as if the estimators of variance components are equal to the true values in the population. The PVA was applied to the V_1 matrix and consequently a more simplified matrix (referred to as the V_2 matrix) was obtained. The elements of the V_2 matrix contained only the true components of variances, and the computational requirement with this matrix was considerably reduced, relative to the V_1 matrix. A further theoretical consideration revealed that the V_2 matrix is equivalent to the inverse of the Fisher information matrix, and therefore this V_2 matrix stands for the large sample asymptotic covariance matrix of the REML estimators. The PVA appeared to be also useful in deriving the large sample (co) variances of the estimators by the REML-like methods such as the pseudoexpectation methods that are not a likelihood-based procedure. For the computation of estimated sampling (co) variances of the REML estimates, however, need to take further theoretical consideration was suggested, since the equivalence found between the derived V_2 matrix and the inverse of the Fisher information matrix implies that the load in the practical calculation of these matrices in which the true values of variances are replaced by the actual REML estimates is still so heavy, particularly in the analysis of a large data set. III. An expression for the average information matrix for the basic mixed linear model and a numerical technique for REML Assuming the basic mixed linear model containing one random effect except for the residual term, an efficient procedure for REML estimation that covers the whole process of the point estimation and the estimation of the large sample sampling (co) variances was developed herein. First, an expression for the average information (AI) matrix was derived. Then, replacing the Hessian matrix by the derived AI matrix, a quasi-Newton type procedure for REML estimation (which will be called the AIREMLm procedure) was defined. The AI matrix derived does not include the vector of the predicted residuals and any incidence matrices, and rather contains a specific type of quadratic form for the subvector of the vector of solutions to the mixed model equations, The quadratic form involved the inverted variance of the random vector and a partitioned matrix of a g-inverse of the mixed model coefficient matrix. With the current AI matrix, however, the calculation of the trace of the matrix products as required in the V_2 matrix or the inverse of the Fisher information matrix was able to be avoided. Thus, an estimate of the large sample (co) variance can be obtained as the inverse of the present AI matrix with true variances replaced by their final estimates, namely the REML estimates. An illustrative numerical analysis using a data-sample of carcass weight in the Japanese Black cattle suggested that the present AIREMLm procedure is superior to the REML procedure using the EM algorithm in terms of the computing time and to the previous procedure using a quasi-Newton method with a different expression for the AI matrix in terms of the computer memory.
IV. An expression for the average information matrix for the general mixed linear model and a numerical technique for REML Assuming the general mixed linear model containing more than two random effects except for the residual effect, the general form of the AI matrix found in the previous chapter was derived. The expanded version of the AIREMLm procedure for estimating multiple components of variances was defined, similarly utilizing the quasi-Newton type equation in which the Hessian matrix was replaced by the derived AI matrix. The current elements of the AI matrix involved a specific type of bilinear form for the vectors of solutions to two different random effects in addition to the quadratic form for each random effect predicted. An illustrative numerical analysis indicated that the current AIREMLm procedure is advantageous over the REML procedure using the EM algorithm and also utilizing the Aitken extrapolation technique in terms of the number of iterations required to attain convergence. V. An integrated numerical technique for REML estimation using the AI and EM algorithms The (quasi-) Newton type method is a procedure commonly used for maximizing nonlinear functions, and is an efficient procedure, as long as successful updating of estimates in the rounds of iteration is achieved. However, with this approach, global convergence is not always guaranteed, and parameter estimates may exceed the boundary. Hence, AIREMLm using a quasi-Newton type method may give negative estimates of variances in some cases. Accordingly, herein, an unified procedure (which will be called AI- REMLme) was proposed, which uses the AI algorithm as its main algorithm, but utilizes the EM estimate to modify the step direction and size, if the procedure fails to increase the function to be maximized. The current procedure, that is AIREMLme, guarantees the converged value within the parameter space. Furthermore, this procedure requires no additional computing time and memory, compared to the AIREMLm procedure, since the first partial derivative of the REML log-likelihood function is already used as the gradient vector in the quasi-Newton iterative equation for the AIREMLm procedure. Results of numerical analyses demonstrated that the present AIREMLme procedure performs very well, even if the AIREMLm procedure fails to give REML estimates, and that convergence with AIREMLme is obviously fast, relative to the EM algorithm using the Aitken extrapolation.
VI. Comparison of computing properties among numerical techniques using the AI, D and DF algorithms for REML estimation To clarify the relative computing property of the AIREMLme procedure, it was compared to certain typical procedures using the AI, D and DF algorithms: the Johnson & Thompson procedure using an AI algorithm, the conventional procedure using the EM algorithm, an accelerated EM procedure, an EM procedure using the Aitken extrapolation and a procedure using the DF algorithm. Two data sets concerning carcass weight of the Japanese Black cattle were analyzed using the two criteria for convergence evaluation. Computing properties compared were the number of iterations, the computing time per round (computing time per likelihood evaluation in the case of the DF algorithm) and the total computing time. All of the computer programs for the AI and D algorithms were written using the FSPAK subroutines, and the 'MTDFREML' program developed by the Animal Research Service, United States Department of Agriculture was utilized for the DF analyses. All procedures gave similar estimates. In details, however, the two AI procedures, or the AIREMLme and the Johnson & Thompson procedures, showed the best numerical property of estimates, while the DF procedure did the poorest one. Also, the AI procedures required the least number of iterations to attain convergence, and the conventional EM procedure necessitated the highest number of iterations. The computing time per round was shortest with the DF approach, while it was longest with the Johnson & Thompson procedure. For the property of the total computing time, AIREMLme was best, and the Johnson & Thompson procedure was a good second, followed by the EM procedure using the Aitken extrapolation, the DF procedure, the accelerated EM procedure and the conventional EM procedure in order of time required. Therefore, further noting that the Johnson & Thompson procedure requires more computer memory than AIREMLme and does not always guarantee convergence within the parameter space, AIREMLme developed herein was considered to be a method superior to the other procedures for REML estimation.
Conclusion The REML method is one of the best procedures for estimating variance components in the general mixed linear model. However, there still exist some important topics to be further addressed on the point and the sampling (co) variance estimation. In this study, noticing the MIVQUE algorithm, a new valid expression for the sampling covariance matrix of the REML estimators was derived, which included the REML estimators in addition to the true values of variances. Because of the heavy computational load with the derived matrix, another simpler matrix for the sampling (co) variances was constructed by proposing an approach based on the large sample theory. The simpler matrix was found to be equivalent to the inverse of the Fisher information matrix which represents the large sample asymptotic sampling (co) variances of the REML estimators, indicating that the computational requirement with the simpler matrix derived is still considerable. Accordingly, to develop an efficient numerical technique for REML estimation that can handle the whole process from the point estimation to the sampling (co) variance estimation, the average information of the REML log-likelihood was noticed, and a new expression for the average information matrix was found. Moreover, a new numerical technique for REML estimation was developed using a quasi-Newton type procedure with the derived AI matrix and incorporating the advantage of the EM algorithm. Being based on a numerical comparison of computing properties among the proposed numerical technique and other typical procedure for REML estimation and considering the functional advantages of the proposed technique, it was concluded that the numerical method developed herein is an efficient, practical and promising procedure for REML estimation of variance components in the general mixed linear model.
学位の種類: 博士（農学）. 報告番号: 甲第1543号. 学位記番号: 新大院博（農）甲第15号. 学位授与年月日: 平成11年3月24日
新大院博（農）甲第15号
新潟大学
1999-03-24
jpn
thesis
http://hdl.handle.net/10191/23703
https://niigata-u.repo.nii.ac.jp/records/4456
1
152
13101A1543
博士（農学）
1999-03-24
新潟大学
https://niigata-u.repo.nii.ac.jp/record/4456/files/000000342001.pdf
application/pdf
16.7 MB
2019-08-05